I found this answer to my question but if the DNS cache for existing.mydomain.com is the problem, wouldn't that solution just result in a infinite circular loop? Or does a DNS lookup for newsub.mydomain.com force a DNS refresh for all subdomains of mydomain.com?

Borealid: The problem is newsub.mydomain.com is being redirected back to existing.mydomain.com (see the answer JWalker posted on that thread). I cannot maintain a parallel newsub.mydomain.com website.
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user51904Aug 21 '10 at 7:30

@Sid Ahuja: I don't think JWalker's solution works. If you're just getting a new IP, simply add both IPs to the server until the TTL expires. If you're actually moving to a new host, replace the old page with a redirect as I said and do all maintenance on the new server. You don't have to maintain a parallel website, just keep the old server sending redirects until your TTL expires.
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BorealidAug 21 '10 at 16:09